1 Chemical Bonds are lines Surface is Electrical Potential Red is positive Blue is negative...

1

Chemical Bonds are linesSurface is Electrical Potential

Red is positiveBlue is negative

Chemist’s View

Ion Channels Proteins with a Hole

All Atoms View

Figure by Raimund Dutzler

~30 Å

2

Channels form a class of Biological Systemsthat can be analyzed with

Physics as Usual

Physics-Mathematics-Engineering are the proper language

for Ion Channelsin my opinion

ION CHANNELS: Proteins with a Hole

3

Physics as Usual

along with

Biology as Usual

Ion Channels can be analyzed with

4

Biology is first of all a Descriptive Science

Biology Involves Many Objects.

The Devices and Machines of Biology must be Named and Described

Then they can be understood

by

Physics as Usual

5

“Why think? ...

Exhaustively experiment.

Then, think”

Claude Bernard

Cited in The Great Influenza, John M. Barry, Viking Penguin Group 2004

Biology as Usual

Biology is first of all a Descriptive Science

Then, Physics as Usual

6

Channels control flow in and out of cells

ION CHANNELS as Biological Objects

~5 µm

7

~30 Å

Ion Channels are the

Main Molecular Controllers “Valves”

of Biological Function

8

Channels control flow of Charged Spheres

Channels have Simple Invariant Structure on the biological time scale.

Why can’t we predict the movement of Charged Spheres through a Hole?

ION CHANNELS as Physical Devices

9

Physical Characteristics of Ion ChannelsNatural Nanodevices

Ion channels have VERY Large Charge Densities critical to I-V characteristics and selectivity (and gating?)

Ion channels have Selectivity. K channel selects K+ over Na+ by ~104.

Ion channels are Device Elements that self-assemble into perfectly reproducible arrays.

Ion channels form Templates for design of bio-devices and biosensors.

Ion channels allow Atomic Scale Mutations that modify conductance, selectivity, and function.

Ion channels Gate/Switch in response to pH, voltage, chemical species and mechanical force

from conducting to nonconducting state.

~30Å


10

Channels Control Macroscopic Flow

with Atomic Resolution

ION CHANNELS as

Technological Objects

11

Ion Channelsare

Important Enough to be

Worth the Effort

12

Goal:Predict FunctionFrom Structure

givenFundamental

Physical Laws

13

Goal, in language of engineers,

Develop Device Equation!

Goal in language of Physiology:Predict Function, from Structure, given Fundamental Physical Laws

14

Current in One Channel Molecule is a Random Telegraph Signal

Voltage Step Applied Here (+ 80 mV; 1M KCl)

Gating: Opening of Porin Trimer

John TangRush Medical Center

16

Single Channel Currents have little variance


17

PLANAR LIPID BILAYER SET UPrecordings on a single molecule!

OmpFtrimer

ions

Trans Cis

OA

Rf

Phospholipid bilayer

-

+

Vcom

Vout

IfIf

IK

IV-converter

Voltage clamp:- voltage is set- current is measured

Functional SurfacesSignalsActuation Lipid Bilayer Setup

Recordings from a Single Molecule

Conflict of Interest

18

Patch clamp and Bilayer apparatus clamp ion concentrations in the baths and the voltage across membranes.

Patch Clamp SetupRecordings from One Molecule

19

Voltage in baths

Concentration in baths

Fixed Charge on channel protein

Current depends on


OmpF KCl 1M 1M||

G119D KCl 1M 1M||

ompF KCl0.05 M

0.05M||

G119D KCl0.05 M

0.05M||

Bilayer Setup

20

Current depends on type of ion

SelectivityJohn TangRush Medical Center

OmpF KCl 1M 1M||

OmpF CaCl2

1M 1M||

Bilayer Setup

21


givenFundamental

Physical Laws

22

Structures…

Location of charges are known with atomic precision (~0.1 Å)

in favorable cases.

23

Charge Mutation in PorinOmpf


Structure determined by x-ray crystallography in Tilman Schirmer’s lab

G119D

24


givenFundamental

Physical Laws

25

But …

What are theFundamental

Physical‘Laws’?

26

Verbal ModelsAre Popular with

Biologistsbut

Inadequate

27

James Clerk Maxwell

“I carefully abstain from asking molecules

where they start…

I only count them….,

avoiding all personal enquiries which would only get me into trouble.”

Royal Society of London, 1879, Archives no. 188In Maxwell on Heat and Statistical Mechanics, Garber, Brush and Everitt, 1995

28

I fear

Biologists use

Verbal Models where

Maxwell abstained

29

Verbal Models are

Vagueand

Difficult to Test

30

Verbal Modelslead to

Interminable Argument and

Interminable Investigation

31

thus,to Interminable Funding

32

and so

Verbal ModelsAre Popular

33

Can Molecular Simulationsserve as

“Fundamental Physical Laws”?

Only if they count correctly !

34

It is very difficult forMolecular Dynamicsto count well enough to reproduce

Conservation Laws (e.g., of energy)

Concentration (i.e., number density) or activity

Energy of Electric FieldOhm’s ‘law’ (in simple situations)

Fick’s ‘law’ (in simple situations)

Fluctuations in number density (e.g., entropy)

35

Can Molecular Simulations Serve as


Simulations are not Mathematics!(e.g., results depend on numerical procedures and round-off error)

Simulations are Reliable Science when they are

Calibrated

36

Simulations are not Mathematics!

Simulations are Reliable Science when they are Calibrated

The computer should not be used in the ‘stand-alone’ mode

quotation from

J.D. Skufca, Analysis Still Matters, SIAM Review 46: 737 (2004)

because results often depend on numerical procedures and round-off error

37

Can Molecular Simulations serve as


Only if Calibrated!

38

Can Molecular Simulations serve as


What should be calibrated?

I believe

Thermodynamics of ions must be calibrated, i.e., activity = free energy per mole,

which means the

Pair Correlation Function

according to classical Stat Mech

39

Calibrated Molecular Dynamics may be possible

MD without Periodic Boundary Conditions

─ HNC HyperNetted Chain

Pair Correlation Function in Bulk Solution

Saraniti Lab, IIT: Aboud, Marreiro, Saraniti & Eisenberg

40

Calibrated Molecular Dynamics may be possible

MD without Periodic Boundary Conditions BioMOCA

─ Equilibrium Monte Carlo (ala physical chemistry)

Pair Correlation Function in Bulk Solution

van der Straaten, Kathawala, Trellakis, Eisenberg & Ravaioli

0.0

0.5

1.0

g+

+(r

)

0.0

0.5

1.0

g--

(r)

EMC

BioMOCA

0

2

4

g+

-(r)

0.0 5.0 15.0 20.0

r ( Å )

10.0

41

Calibrated MD may be possible, even in aGramicidin channel

Molecular Dynamics without Periodic Boundary Conditions BioMOCA

van der Straaten, Kathawala, Trellakis, Eisenberg & Ravaioli

0 1 2 3 4 5 6 7 8

Channel current (pA)

0

2

4

6

8

Cou

nt

Simulations 235ns to 300ns, totaling 4.3 μs.Mean I = 3.85 pA, 24 Na+ crossings per 1 μs

16 Na+ single channel currents

42

Essence of Engineering is knowing

What Variables to Ignore!

WC Randels quoted in Warner IEEE Trans CT 48:2457 (2001)

Until Mathematics of Simulations is available we take an

Engineering Approach

43

What variables should we ignorewhen we make low resolution models?

How can we tell when a model is helpful?Use the scientific method

Guess and Check!

Intelligent Guesses are MUCH more efficient

Sequence of unintelligent guesses may not converge! (e.g., Rate/State theory of channels/proteins)

44

Guess Cleverly!

Without qualitative understanding,quantitative models can be only

vague statements of thermodynamics.

Qualitative Simulationshave an important role in

Computational Biology

45

Goal in language of Physiology:Predict Function, from Structure, given Fundamental Physical Laws

Goal, in language of engineers,

Develop Device Equation!

46

Use

Theory of Inverse Problems

(Reverse Engineering)

optimizes “Guess and Check”

1) Measure only what can be measured

(e.g., not two resistors in parallel).

2) Measure what determines important parameters

3) Use efficient estimators.

4) Use estimators with known bias

5) no matter what the theory,

Guess Cleverly!

47

Use the scientific methodGuess and Check!

When theory works, need few checks Computations (almost) equal experiments

Structural EngineeringCircuit Design

Airplane DesignComputer Design

Can be done (almost) by theory,

48

Channels are only Holes

Why can’t we have a fully successful theory?Must know physical basis to make a good theory

Physical Basis of Gating is not known

Is the physical basis of permeation known?

49

Proteins Bristle with Charge Cohn (1920’s) & Edsall (1940’s)

Ion Channels are no exception

We start with

Electrostaticsbecause of biology

50−0.55O

0.35HH_22

0.60C

0.50C_z

0.35HH_21

−0.45NH_2

0.35HH_12

−0.45NH_1

0.35HH_11

0.32Average Magnitude

0.30H_e

−0.40N_e

0.10C_d

0.00C_g

0.00C_b

0.10C_a

0.25H

−0.40N

Charge (units /e)Atom

Atom in Arginine Charge

according to CHARMM

51

Many atoms in a protein have

Permanent Charge ~1e

Permanent charge is the (partial) charge on the atom when the local electric field is zero.

Active Sites in Proteins have

Many Charges in a Small Place

52

Active Sites of Proteins are Very Charged e.g. 7 charges ~ 20 M net charge

Selectivity Filters and Gates of Ion Channels are

Active Sites

1 nM

= 1.2×1022 cm-3

53

in a Sphere of diameter 14 Å

Small Charge in Small Places

Large Potentials*

Start with electrostatics because

*Current varies 2.7× for 25 mV because thermal energy = 25 meV

1 charge gives

1

12.5

V at psec times; dielectric constant = 2

mV at µsec times; dielectric constant = 80

54

We start without quantum chemistry*(only at first)

* i.e., orbital delocalization

55

Although we start with electrostatics

We will soon add

Physical Models of

Chemical Effects

56

It is appropriate to be skepticalof analysis in which the only chemistry is physical

But give me a chance, ask

Or email [email protected]

for the papers

57

Very Different from Traditional Structural Biology

which, more or less,

ignores Electrostatics

58

Poisson-Nernst-Planck (PNP)

Note: physicists rarely ignore the electric field, even to begin with.

Lowest resolution theory that includes

Electrostatics and Flux is (probably)

PNP, Gouy-Chapman, (nonlinear) Poisson-Boltzmann, Debye-Hückel, are fraternal twins or siblings with similar resolution

59

ION CHANNEL MODELS

1 ;i i i iD

kT J x x x x

Poisson’s Equation

Drift-diffusion & Continuity Equation

0 i i

i

e z x x x

Stay Tuned … Derivation on the way: Schuss, Singer, Nadler et al

0; J x

Chemical Potential

ex*

ln ii i iz e kT

xx x x

Special Chemistry

60

One Dimensional PNP

ii i i

dJ D x A x x

dx

Poisson’s Equation

Drift-diffusion & Continuity Equation

0

i i

i

d dx A x e z x

A x dx dx

0idJ

dx

Chemical Potential

ex*ln i

i i iz e kT

xx x x

Special Chemistry

61

Poisson Equation and

Nernst Planck Equation (Fick’s Law for charged particles)

are solved together by

Gummel iterationPoisson Transport Poisson Transport

Or much better(but much harder)

Newton Iteration

62

I

V

INPUT OUTPUTTHEORY

PNP

STRUCTURE

Length, diameter,

Permanent Charge

Dielectric Coefficient

EXPERIMENTAL CONDITIONS

Bath concentrations

Bath Potential Difference

PNP Forward Problem

63

Current Voltage Relation Gramicidin 3D PNP Uwe Hollerbach

64

Fit of 1D PNP Current-Voltage 100 mM KCl

OmpFG119D

Duan ChenJohn TangRush Medical Center

65

V

I

PNP -1 Inverse Problem

Input OutputTheory

EXPERIMENTAL CONDITIONS

Bath Concentrations

Bath Potential Difference

EXPERIMENTS

ELECTRICAL STRUCTURE

Length, diameter,

Permanent Charge

Dielectric Coefficient

PNP -1

Inverse Problem for MathematiciansWhat measurements

best determine electrical structure?

Stay tuned, Berger, Engl & Eisenberg

66

Charge Mutation in PorinOmpf G119D


Structure determined by x-ray crystallography in Tilman Schirmer’s lab

67

Net Charge Difference= 0.13 1.1= 0.97e

Duan ChenJohn Tang

68

Shielding DominatesElectric Properties of Channels,

Proteins, as it does Ionic Solutions

Shielding is ignored in traditional treatments

of Ion Channels and of Active Sites of proteins

Rate Constants Depend on Shielding and so

Rate Constants Depend on Concentration and Charge

Main Qualitative Result

69

Main Qualitative ResultShielding in Gramicidin

Uwe HollerbachRush Medical Center

70

PNP misfits in some cases

even with ‘optimal’ nonuniform D(x)

Duan ChenJohn TangRush Medical Center

71

Shielding/PNP is not enoughPNP includes Correlations

only in the Mean Field

PNP ignores ion- ion correlations

and

discrete particle effects:

Single Filing, Crowded Charge

Dielectric Boundary Force

72

Neither Field Theory nor Statistical Mechanics

easily accommodates Finite Size

of Ions and Protein Side-chainsHow can that be changed?

Learn from Mathematicians

and/or

Physical Chemists

Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!

73

Learning with MathematiciansZeev Schuss, Boaz Nadler, Amit Singer

Dept’s of MathematicsTel Aviv University, Yale University

Molecular Biophysics, Rush Medical College

We extend usual chemical treatment to include flux and spatially nonuniform boundary conditions

We have concrete results only in the uncorrelated case!

We have learned how to derive PNP (by mathematics alone).

Count trajectories, not states.Compute Field Equations,

not pair-wise forces.

74

Counting Langevin Trajectories in a Channel

(between absorbing boundary conditions)

implies PNP (with some differences)

PNP measures the density of trajectories (nearly)

Zeev Schuss, Amit Singer: Tel Aviv UnivBoaz Nadler: Yale Univ,

75

Conditional PNP

0 |

| |

yy y perme

p n

y x y

c y x c y x

Electric Force depends on Conditional Density of Charge

Nernst-Planck gives UNconditional Density of Charge

1

| 0x p y y xc x e y x DBF

m x

massfriction

Schuss, Nadler, Eisenberg

76

Boaz Nadler and Uwe HollerbachYale University Dept of Mathematics & Rush Medical Center

Dielectric Boundary ForceDBF

0

04

qq

0

0 r r

r

r rF r r

PNP ignores correlations induced by

Discovered by Coalson-Kurnikova, Chung-Corry, … not us!

Dielectric Boundary Force for point charge 0F r q r

77

Counting gives PNP with corrections

1) Correction for Dielectric Boundary Force*

2) NP (diffusion) equation describes probability of location but

3) Poisson equation depends on a conditional probability of location of charge, i.e., probability of location of trajectories, given that a positive ion is in a definite location.

4) Relation of is not known but can be estimated by closure relations, as in equilibrium statistical mechanics.

5) Closure relations involve correlations from single filing, finite volume of ions, boundary conditions, not well understood, … yet. Stay Tuned!

0 |r r

P

r

P

0 |r r r

P Pand

*by derivation, not assumption

78

Until mathematics is available, we

Follow the Physical Chemists, even if

their approximations are ‘irrational’, i.e., do not have error bounds.

Bob Eisenberg blames only himself for this approach

79

Physical Chemistry has shown that

Chemically Specific Properties of ions

come from their

Diameter and Charge (much) more than anything else.

Physical Models are Enough


80

Physical Theories of Plasma of Ions

Determine (±1-2%) Activity* of ionic solutions

from

Infinite dilution,to

Saturated solutions, even in

Ionic melts.

*Free Energy per Mole


81Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!

Ionsin a solution are a

Highly Compressible Plasma

Central Result of Physical Chemistry

although the

Solution itself is

Incompressible

82

1 ( ) ( ) ( )i i i ikT D J x x x}

Concentration-independent• geometric restrictions• solvation (Born) terms

Ideal term• electrochemical potential of point particles in the

electrostatic mean-field• includes Poisson equation

Excess chemical potential•Finite size effects•Spatial correlations

Chemical potential has three components

0 id ex( ) ( ) ( ) i i ix x x

83

Simplest Physical Theory MSA

1 2

2 2 2 20

0

Electrochemical Potential of species

ln

24 1 1 3

i

HS ESi i i i

HS i i i ii i

i i

i

z F RT

e z z

Ideal Excess

Electrostatic SpheresHard Spheres

Ionic radii I are known

values in bulk

Learned from Lesser Blum & Doug Henderson … Thanks! We use Simonin-Turq formulation.

84

Properties of Highly Compressible Plasma of Ions

Similar Results are computed by many

Different theories and Simulations

MSA is only simplest.

We (and others) have used MSA, SPM, MC, and DFT

MSA: Mean Spherical ApproximationSPM: Solvent Primitive ModelMC: Monte Carlo Simulation

DFT: Density Functional Theory of Solutions

Most Accurate

Atomic Detail

Calibrated !

Inhomogeneous Systems

85

back to channels

Selectivity in Channels

Wolfgang Nonner, Dirk GillespieUniversity of Miami and Rush Medical Center

86

Binding Curve

Wolfgang Nonner

87

O½

Selectivity Filter

Selectivity Filter Crowded with Charge

Wolfgang Nonner

MSA Theory of Selective BindingClassical Donnan Equilibrium of Ion Exchanger

Solve the simultaneous equations for by iterationiμ

Protein Permanent Charge

Volume Occupied by Protein

Pressures are not Equal

Mobile Ions: Mobile Ionsout ini iμ μ

89

Understand Selectivity well enough to

Make a Calcium Channelusing techniques of molecular genetics,

site-directed Mutagenesis

Goal:

George Robillard, Henk Mediema, Wim Meijberg

90

General Biological Theme (‘adaptation’)Selectivity Arises in a Crowded Space

Biological case:0.1 M NaCl and 1 M CaCl2

in the baths

As the volume is decreased,water is excluded from the filter by crowded charge effectsCa2+ enters the filter and displaces Na+

Biological Case

Wolfgang Nonner Dirk Gillespie

91

Sensitivity to Parameters

VolumeDielectric

Coefficient

92

Trade-offs ‘1.5’ adjustable parameters

Volume

Die

lect

ric

Co

effi

cien

t

93

Binding CurvesSensitivity to Parameters

0.75 nm3 volume 0.20 nm3 volume

94

At Large Volumes

Electrical Potential can Reverse0.75 nm3 volume 0.20 nm3 volume

- - 0 Potential- -

0 Potential

Positive

Negative

Negative

95

Competition of Metal Ions vs. Ca++ in L-type Ca Channel

Nonner & Eisenberg

96

Similar Results have been found by

Henderson, Boda, et al.Hansen, Melchiona, Allen, et al.,

Nonner, Gillespie, Eisenberg, et al.,Using MSA, SPM, MC and DFT

for the L-type Ca Channel



Atomic Detail

Calibrated!

Most Accurate


97

Best Result to Date with Atomic Detail Monte Carlo,

including Dielectric Boundary Force

Dezso Boda, Dirk Gillespie, Doug Henderson, Wolfgang Nonner

Na+

Na+

Ca++

Ca++

98

Other

Properties of Ion Channels are likely to involve

more subtle physics including

orbital delocalizationand

chemical binding Selectivity apparently does not!


99

Ionic Selectivity in Protein ChannelsCrowded Charge Mechanism

Simplest Version: MSA

How doesCrowded Charge give Selectivity?

100

General Biological Theme (‘adaptation’)Selectivity Arises in a Crowded Space

Biological case:0.1 M NaCl and 1 M CaCl2

in the baths

As the volume is decreased,water is excluded from the filter by crowded charge effectsCa2+ enters the filter and displaces Na+

Biological Case


101


4 Negative Charges of glutamates of protein

DEMAND 4 Positive Charges

nearby

either 4 Na+ or 2 Ca++

102


Simplest Version: MSA

2 Ca++ are LESS CROWDED than 4 Na+,

Ca++ SHIELDS BETTER than Na+, so

Protein Prefers Calcium

1032 Ca++ are LESS CROWDED than 4 Na+

104

What does the protein do?

Selectivity arises from Electrostatics and Crowding of Charge

Certain MEASURES of structure are

Powerful DETERMINANTS of Functione.g., Volume, Dielectric Coefficient, etc.

Precise Arrangement of Atoms is not involved in the model, to first order.

105

Protein provides Mechanical Strength

Volume of PoreDielectric Coefficient/Boundary

Permanent Charge

Precise Arrangement of Atoms is not involved in the model, to first order.

but

Particular properties ‘measures’ of the protein are crucial!

What does the protein do?

106

Mechanical StrengthVolume of Pore

Dielectric Coefficient/BoundaryPermanent Charge

But not the precise arrangement of atoms

Implications for Artificial Channels

Design Goals are

107

Implications for Traditional Biochemistry

Traditional Biochemistry focuses on

Particular locations of atoms

108

Traditional Biochemistry assumes

Rate Constants Independent

of Concentration & Conditions

109

Implications for Traditional Biochemistry

Traditional Biochemistry(more or less)

Ignores the Electric Field

110

ButRate Constants depend steeply

on

Concentration and

Electrical Properties*

because of shielding, a fundamental property of matter, independent of model, in my opinion.

*nearly always

111

Electrostatic Contribution to ‘Dissociation Constant’ is large

and is an

Important Determinant of Biological Properties

Change of Dissociation ‘Constant’

with concentration is large and is an

Important Determinant of Biological Properties

112

Traditional Biochemistryignores

Shielding and Crowded Charge although

Shielding DominatesProperties of Ionic Solutions

and cannot be ignored in Channels and Proteins

in my opinion

113



How can we use these ideas?


BioMaDe Corporation, Groningen, Netherlands

114

More?

115

Functioncan be predictedFrom Structure

givenFundamental

Physical Laws(sometimes, in some cases).

Conclusion

116

More?

117



How can we use these ideas?


BioMaDe Corporation, Groningen, Netherlands

118

Strategy

Use site-directed mutagenesis to put in extra glutamates

and create an EEEE locus in the selectivity filter of OmpF

Site-directed

mutagenesis

R132

R82E42

E132

R42 A82

Wild type EAE mutant

E117 E117

D113D113

George Robillard, Henk Mediema, Wim MeijbergBioMaDe Corporation, Groningen, Netherlands

119

-100 -50 50 100

-150

-50

50

150

ECa

WT

EAE

Current (pA)

Voltage (mV)

Cis Trans

1 M CaCl2 0.1 M CaCl2

Ca2+

Ca2+

IV-PLOT

Cis Trans Cis Trans

IV-plot EAE: current reverses at equilibrium potential of Ca2+ (ECa),

indicating the channel can discriminate between Ca2+ and Cl-

Zero-current potentialor reversal potential = measure of ion selectivity

Henk MediemaWim Meijberg

120

PCa/PCl

WT 2.8AAA 25EAE >100

Ca2+ over Cl- selectivity (PCa/PCl)recorded in 1 : 0.1 M CaCl2

SUMMARY OF RESULTS (1)

Conclusions:

- Taking positive charge out of the constriction zone (= -3, see control mutant AAA) enhances the cation over anion permeability.

- Putting in extra negative charge (= -5, see EAE mutant) further increases the cation selectivity.


121

PCa/PNa

WT 2.2AAA 3.7EAE 4.2

Ca2+ over Na+ selectivity (PCa/PNa)recorded in 0.1 M NaCl : 0.1 M CaCl2

SUMMARY OF RESULTS (2)

Conclusion:

- Compared to WT, EAE shows just a moderate increase of the Ca2+ over Na+ selectivity.

- To further enhance PCa/PNa may require additional negative charge and/or a change of the ‘dielectric volume’.

Work in Progress!


122

Selectivity Differs

in Different Types of ChannelsWolfgang Nonner

Dirk Gillespie

Other Types of Channels

123

CaCa channelchannel Na channelNa channel Cl channelCl channel K channelK channel

prefers

Small ions Ca2+ > Na+

prefers

Small ions Na+ > Ca2+

Na+ over K+

prefers

Large ionsprefers

K+ > Na+

Selectivity filter

EEEE 4 − charges

Selectivity filter

DEKA2 −, 1+ charge

Selectivity filter

hydrophobicpartial charges

Selectivity filter

single filingpartial charges

PNP/DFT Monte Carlo Bulk Approx Not modeled yet

Selectivity of Different Channel Types

The same crowded charge mechanism can explain allthese different channel properties

with surprisingly little extra physics .

124

Sodium Channel

(with D. Boda, D. Busath, and D. Henderson)

Related to Ca++ channel• removing the positive lysine (K) from the DEKA locus makes calcium-selective channel

High Na+ selectivity• 1 mM CaCl2 in 0.1 M NaCl gives all Na+ current (compare to

calcium channel)• only >10 mM CaCl2 gives substantial Ca++ current

Monte Carlo method is limited (so far) to a uniform dielectric

Stay tuned….


125

Na+

Ca++

Na+/Ca2+ Competition in the Sodium Channel

Biological Region

Ca++ in bath (M)


126

•Model gives small-ion selectivity. •Result also applies to the calcium channel.

Na+/Alkali Metal Competition in Na+ Channel

Biological Region


127

‘New’ result from PNP/SPM combined analysis

Spatial Nonuniformity in Na+ Channel


128

Na+ vs K+ Selectivity

Na+

K+ Channel

Protein

ProteinProtein

Protein

Na+ Channel


129

Na+ Channels Select Small Na+ over Big K+

because(we predict)

Protein side chains are small

allowing

Small Na+ to Pack into Niches

K+ is too big for the niches!Wolfgang Nonner Dirk Gillespie

Summary of Na+ Channel

130

Sodium Channel Summary

Na+ channel is a Poorly Selective

Highly Conducting Calcium channel,

which is Roughened so it prefers

Small Na+ over big K+


131

Cl− Selective ChannelSelective for Larger Anions

The Dilute ChannelWolfgang Nonner Dirk Gillespie

132

Chloride Channel

Channel prefers large anions in experiments,

Low Density of Charge (several partial charges in 0.75

nm3)

Selectivity Filter contains hydrophobic groups

• these are modeled to (slightly) repel water• this results in large-ion selectivity

Conducts only anions at low concentrations

• Conducts both anions and cations at high concentration

Current depends on anion type and concentrationWolfgang NonnerDirk GillespieDoug HendersonDezso Boda

133

Biological Case

Chloride ChannelSelectivity depends Qualitatively on concentration


134

The Dilute Channel: Anion Selective

Channel protein creates a

Pressure difference between bath and channel

Large ions like Cl– are Pushed into the channel more than smaller ions

like F–


135

Key: Hydrophobic Residues Repel Water giving• Large-ion selectivity

(in both anion and cation channels). • Peculiar non-monotonic conductance properties

and IV curves observed in experiments

Hydrophobic repulsion can give gating. ‘Vacuum lock’ model of gating

(M. Green, D. Henderson; J.-P. Hansen; Mark Sansom; Sergei Sukarev)

Chloride Channel


136

Conclusion

Each channel type is a variation on a theme of

Crowded Chargeand Electrostatics,

Each channel types uses particular physics as a variation.


137

Functioncan be predictedFrom Structure

givenFundamental

Physical Laws(sometimes, in some cases).

138

More?

DFT

139

Density Functionaland

Poisson Nernst Planckmodel of

Ion Selectivity in

Biological Ion Channels

Dirk GillespieWolfgang Nonner

Department of Physiology and BiophysicsUniversity of Miami School of Medicine

Bob EisenbergDepartment of Molecular Biophysics and Physiology

Rush Medical College, Chicago

140

We (following many others) have used

Many Theories of Ionic Solutions

asHighly Compressible Plasma of Ions

with similar results

MSA, SPM, MC and DFT



Most Accurate

Atomic Detail


141

Density Functionaland

Poisson Nernst Planckmodel of

Ion Selectivity in

Biological Ion Channels

Dirk GillespieWolfgang Nonner

Department of Physiology and BiophysicsUniversity of Miami School of Medicine

Bob EisenbergDepartment of Molecular Biophysics and Physiology

Rush Medical College, Chicago

142

We (following many others) have used

Many Theories of Ionic Solutions

asHighly Compressible Plasma of Ions

with similar results

MSA, SPM, MC and DFT



Most Accurate

Atomic Detail


143

Density Functional Theory

HS excess chemical potential is from free energy functional

HS k HS n d x x x

Energy density depends on “non-local densities”

Nonner, Gillespie, Eisenberg

144

HS kHSi

i

HSikT n d

n

xx

x

x x x x

HS excess chemical potential is

Free energy functional is due to Yasha Rosenfeld and is considered more than adequate by most physical chemists.

The double convolution is hard to compute efficiently.


We have extended the functional to

Charged Inhomogeneous Systems with a bootstrap perturbation method

that fits MC simulations nearly perfectly.

145

Example of an Inhomogeneous Liquid

A two-component hard-sphere fluid near a wall in equilibrium (a small and a large species).

Near the wall there are excluded-volume effects that cause the particles to pack in layers.

These effects are very nonlinear and are amplified in channels because of the high densities.

small species

large species


146

The ProblemWe are interested in computing the flux of ions between two baths of fixed ionic concentrations. Across the system an electrostatic potential is applied.

Separating the two baths is a lipid membrane containing an ion channel.

ionic concentrations and electrostatic potentialheld constant far from

channel

ionic concentrations and electrostatic potentialheld constant far from

channel

membrane with ion channel


147

Modeling Ion Flux

The flux of ion species i is given by the constitutive relationship

1i i i iD

kT J x x x

The flux follows the gradient of the total chemical potential.

where Di is the diffusion coefficienti is the number densityi is the total chemical potential of species i


148

1 ( ) ( ) ( )i i i ikT D J x x x}

0 id ex( ) ( ) ( ) i i ix x x

concentration-independent• geometric restrictions• solvation (Born) terms

ideal term• electrochemical potential of point particles

in the electrostatic mean-field• includes Poisson equation

excess chemical potentialthe “rest”: the difference between the “real” solution and the ideal solution

The chemical potential has three components


149

When ions are charged, hard spheres the excess chemical potential is split into two parts

1 ( ) ( ) ( )i i i ikT D J x x x}

0 id ex( ) ( ) ( )i i i x x x}HS ES( ) ( )i i x x

Electrostatic Componentdescribes the electrostatic effects of charging up the ions

Hard-Sphere Componentdescribes the effects of excluded volume

the centers of two hard spheres of radius R cannot come closer than 2R


150


HS excess chemical potential comes from free energy functional

HS k HS n d x x x

Energy density depends on “non-local densities”


151

HS kHSi

i

HSikT n d

n

xx

x

x x x x

HS excess chemical potential is

Free energy functional is due to Yasha Rosenfeld and is considered more than adequate by most physical chemists.

The double convolution is hard to compute efficiently.


We have extended the functional to

Charged Inhomogeneous Systems with a bootstrap perturbation method

that fits MC simulations nearly perfectly.

152


Energy density depends on “non-local densities”:

1 2 1 20 3

3

332 22

2 223

ln 11

124 1

V VHS

V V

n nn n n

n

n

nn

n nx

n n


153Nonner, Gillespie, Eisenberg

The non-local densities ( = 0, 1, 2, 3, V1, V2) are averages of the local densities:

2 3

0 1 22

2 1 2

4 4

4

i i

i

i i i i

i i i i i

V V Vi i i i i

n d

R R

R R

R R

x x x x x

r r r r

r r r

rr r r r

r

1

where is the Dirac delta function, is the Heaviside step

function, and Ri is the radius of species i. 1

154

We use Rosenfeld’s perturbation approach to compute the electrostatic component.

Specifically, we assume that the local density i(x) is a perturbation of a reference density iref(x):

i x i

ref x i x

The ES Excess Chemical PotentialDensity Functional Theory


155

The Reference FluidIn previous implementations, the reference fluid was chosen to be a bulk fluid. This was both appropriate for the problem being solved and made computing its ES excess chemical potential straight-forward.

However, for channels a bulk reference fluid is not sufficient. The channel interior can be highly-charged and so 20+ molar ion concentrations can result. That is, the ion concentrations inside the channel can be several orders of magnitude larger than the bath concentrations.

For this reason we developed a formulation of the ES functional that could account for

such large concentration differences.


156

Test of ES FunctionalTo test our ES functional, we considered an equilibrium problem designed to mimic a calcium channel.

two compartments were equilibratededge effects fully computed

24 M O-1/2

CaCl2

NaClorKCl

0.1 M

The dielectric constant was 78.4 throughout the system.


157


158


159

Conclusion:

Density Functional Theory can

Include Electrostatics

160

‘New’ Mathematics is Needed:

Analysis of Simulations

161

Can Simulations serve as


Direct Simulations are Problematic Even today

162

Can simulations serve as fundamental physical laws?

Direct Simulations are Problematic Even today

Simulations so far cannot reproduce macroscopic variables and phenomena known

to dominate biology

163

Simulations so far often do not reproduce

Concentration (i.e., number density)

(or activity coefficient) Energy of Electric Field Ohm’s ‘law’ (in simple situations)

Fick’s ‘law’ (in simple situations)

Conservation Laws (e.g., of energy)

Fluctuations in number density

164

The larger the calculation, the more work done, the greater the error

First Principle of Numerical Integration

First Principle of Experimentation

The more work done, the less the error

Simulations as fundamental physical laws (?)

165

How do we include Macroscopic Variables in

Atomic Detail Calculations?

Another viable approachis

Hierarchy of Symplectic Simulations

166

Analysis of Simulations

e.g.,How do we include

Macroscopic VariablesConservation laws

in

Atomic Detail Calculations?

Because mathematical answer is unknown, I use an Engineering Approach

Hierarchy of Low Resolution Models

167

Simulations produce too many numbers

106 trajectories each 10-6 sec long, with 109 samples in each trajectory,

in background of 1022 atoms

Why not simulate?

168

Simulations need a theory that

Estimates Parameters (e.g., averages)

or

Ignores Variables

Theories and Models are Unavoidable! (in my opinion)

169

Symplectic integrators are precise in‘one’ variable at a time!

It is not clear (at least to me) that symplectic integrators can be precise in all relevant variables at one time

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